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Wild randomness and globality


The River Nile and Infinite Systems
Benoît Mandelbrot Mathematician
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Let me perhaps mention about the River Nile. I heard about the River Nile about the same time as I heard about turbulence, in '64, and immediately had a little model for it, which- well, was very strange in some ways. The River Nile had many features, the flows of the Nile over the years have many features that the physicists would later have called anomalies. You evaluate the variance, and the variance doesn't draw like time span as you take bigger and bigger integration intervals; it goes like time to a certain power different from one. And then a man named H. E. Hurst introduced a technique, which I later sort of formalised as R/S; he was getting results that were incomprehensible. This was very much noticed and the literature grew about it. Many people of considerable talent said that this required just a little bit more fiddling with microprocessors or a bit more memory, or it will require some rethinking, but it was viewed as a major puzzle, this thing which didn't work out. Well one day, William Fellow, who was up and coming at IBM as a consultant, was in my office and, and so I asked him, "You wrote this paper on Hurst's findings; you say that one could do this and that - but I think that wouldn't work. As a matter of fact, I know how to make it work." He says, "Impossible. There's no sensible way of doing it." I said, "Well, there's a sensible way here.... " "Of course, of course, of course." Once this barrier of infinite dependence was overthrown, which was a colossal barrier, the answer was obvious, but you had to accept the notion that there is infinite dependence between what's happening now and what's happening way in the past and way in the future. Now, it's a very difficult idea to accept. The consequences of this fundamental idea are hard to accept and it's - When you say infinite dependence, this means a power-law dependence? A power-law dependence, yes. I mean it's not infinity, the size; it is still the case, but bears the effect on the long periods. That's right. And many people in many contexts have been arguing strongly against it, and since it's a very important feature let me say a few words about it. When it became quite clear that to analyse something like the River Nile over a thousand years, or some other data over many thousands of years, or IBM stock over thirty-five years - which still represents a very large number of thousands of data, if infinite dependence is necessary it does not mean that IBM's details of ten years ago influence IBM today, because there's no mechanism within IBM for this dependence. However, IBM is not alone. The River Nile is alone. They're just one-dimensional corners of immensely big systems.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Daniel Zajdenweber Bernard Sapoval

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Duration: 3 minutes, 23 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008